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Find the domain of the real-valued function

f(x) = -sqrt(-10x^2 - 4x + 6)

 

Give the endpoints in your answer as common fractions, not mixed numbers or decimals.

 Mar 12, 2021
 #1
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f(x)  =  - sqrt ( -10x^2  - 4x  + 6)

 

We must  have  that

 

-10x^2  - 4x + 6  ≥  0                multiply through  by  -1  and reverse the inequality sign

 

10x^2  + 4x  - 6   ≤  0                   factor as

 

(10x  - 6) ( x + 1)     ≤  0          (1)

 

10x  - 6  =  0                        x + 1   = 0

10x  = 6                               x  =  -1

x =  6/10  =  3/5

 

We  have  three intervals to consider    (-inf, -1] , [ -1, 3/5 ]  , [ 3/5, +inf )

 

The midle  interval is the one that  makes  (1)  true

 

So.....the domain is   [ -1, 3/5 ]

 

See the graph here :  https://www.desmos.com/calculator/xwiv35snre

 

 

cool cool cool

 Mar 12, 2021

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